Celebratio Mathematica

M. F. Atiyah and F. Hirzebruch: “Riemann–Roch theorems for differentiable manifolds,” Bull. Am. Math. Soc. 65 : 4 (1959), pp. 276–281. MR 110106 Zbl 0142.40901 article

The Riemann–Roch Theorem for an algebraic variety \( Y \) (see [Hirzebruch 1956]) led to certain divisibility conditions for the Chern classes of \( Y \). It was natural to ask whether these conditions held more generally for any compact almost complex manifold. This question, and various generalizations of it, were raised in [Hirzebruch 1954] and most of these have since been answered in the affirmative in [Borel and Hirzebruch 1958] and [Milnor 1960].

More recently Grothendieck has obtained [Borel and Serre 1958] a more general Riemann–Roch Theorem for a map \( f: Y\to X \) of algebraic varieties. This reduces to the previous Riemann–Roch Theorem on taking \( X \) to be a point. Grothendieck’s Theorem implies many conditions on characteristic classes, and again it is natural to ask if these conditions hold more generally for almost complex or even differentiable manifolds. The purpose of this note is to enunciate certain differentiable analogues of Grothendieck’s Theorem. These “differentiable Riemann–Roch Theorems” yield, as special cases, the divisibility conditions mentioned above and also certain new homotopy invariance properties of Pontrjagin classes. As an application of the latter we get a new proof (and slight improvement) of the result of Kervaire–Milnor [1960] on the stable \( J \)-homomorphism.

Another differentiable Riemann–Roch Theorem, with applications to embeddability problems of differentiable manifolds, will be found in [Atiyah and Hirzebruch 1959].

The proofs of our theorems rely heavily on the Bott periodicity of the classical groups [Bott 1957, 1958, 1959], and are altogether different from the earlier methods of [Borel and Hirzebruch 1958] and [Milnor 1960], which were based on Thom’s cobordism theory and Adams’ spectral sequence.

M. F. Atiyah and F. Hirzebruch: “Quelques théorèmes de non-plongement pour les variétés différentiables” [Some non-immersion theorems for differentiable manifolds], Bull. Soc. Math. France 87 (1959), pp. 383–396. MR 114231 Zbl 0196.55903 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958], [Borel and Hirzebruch 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espace euclidien (ou ce qui est équivalent, une sphère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui étaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott [Borel and Hirzebruch 1958/59; Bott 1958] qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

M. F. Atiyah and F. Hirzebruch: “Quelques théoremes de non-plongement pour les variétés différentiables” [Some non-embedding theorems for differentiable manifolds], Colloques Int. Centre Nat. Rech. Sci. 89 (1960), pp. 383–396. See also Bull. Soc. Math. France 87 (1959). Zbl 0108.18202 article

Nous avons montré dans [Atiyah and Hirzebruch 1959] que le théorème de Riemann–Roch [Borel and Serre 1958] a des analogues différentiables. Un exposé de ces résultats a été fait par l’un des auteurs au Séminaire Bourbaki [Hirzebruch, 1958/59]. Les théorèmes de Riemann–Roch différentiables fournissent comme cas particulier certaines conditions de divisibilité pour les classes caractéristiques d’une variété différentiable que l’on peut considérer comme des analogues différentiables du théorème de Riemann–Roch de [Hirzebruch 1956].

La plupart de ces conditions de divisibilité ont été prouvées précédement dans [Borel and Hirzebruch 1958; 1960] et [Milnor 1960]. Dans ce qui suit nous démontrons à l’aide des méthodes de [Atiyah and Hirzebruch 1959] que les classes caractéristiques d’une variété différentiable compacte orientée de dimension \( d \) satisfont aux conditions de divisibilité supplémentaires si la variété peut être différentiablement plongée dans un espance euclidien (ou ce qui est équivalent, une spère) de dimension \( 2d - q \). Ces conditions de divisibilité «non stables» nous permettent de prouver des théorèmes de non-plongement qui semblent beaucoup plus forts que ceux qui ’\etaient connus avant (3.6). L’outil essentiel est encore le théorème de Bott ([Borel and Hirzebruch, 1958/59] et [Bott 1958]) qui dit que la \( n \)-ième classe de Chern d’un fibré vectoriel complexe sur la sphère \( S_{2n} \) est divisible par \( (n-1)! \).

M. F. Atiyah and F. Hirzebruch: “Vector bundles and homogeneous spaces,” pp. 7–38 in Differential geometry (Tucson, AZ, 18–19 February 1960). Edited by C. B. Allendoerfer. Proceedings of Symposia in Pure Mathematics 3. American Mathematical Society (Providence, RI), 1961. Republished in Algebraic topology: A student’s guide (1972). MR 139181 Zbl 0108.17705 incollection

F. Hirzebruch	Related	Volume
Carl Barnett Allendoerfer	Related

M. F. Atiyah and F. Hirzebruch: “Cohomologie-Operationen und charakteristische Klassen” [Cohomology operations and characteristic classes], Math. Z. 77 : 1 (1961), pp. 149–187. Dedicated to Friedrich Karl Schmidt on the occasion of his sixtieth birthday. MR 156361 Zbl 0109.16002 article

F. Hirzebruch	Related	Volume
Friedrich Karl Schmidt	Related

M. F. Atiyah and F. Hirzebruch: “Bott periodicity and the parallelizability of the spheres,” Proc. Camb. Philos. Soc. 57 : 2 (April 1961), pp. 223–226. MR 126282 Zbl 0108.35902 article

The theorems of Bott [1958, 1959a] on the stable homotopy of the classical groups imply that the sphere \( S^n \) is not parallelizable for \( n \neq 1 \), \( 3, 7 \). This was shown independently by Kervaire [1958] and Milnor [1958; Bott and Milnor 1958]. Another proof can be found in [Borel and Hirzebruch 1959, §26.11]. The work of J. F. Adams (on the non-existence of elements of Hopf invariant one) implies more strongly that \( S^n \) with any (perhaps extraordinary) differentiable structure is not parallelizable if \( n \neq 1 \), \( 3, 7 \). Thus there exist already four proofs for the non-parallelizability of the spheres, the first three mentioned relying on the Bott theory, as given in [Bott 1958, 1959a]. The purpose of this note is to show how the refined form of Bott’s results given in [Bott 1959b] leads to a very simple proof of the non-parallelizability (only for the usual differentiable structures of the spheres). We shall prove in fact the folowing theorem due to Milnor [1958] which implies the non-parallelizability.

There exists a real vector bundle \( \xi \) over the sphere \( S^n \) with \( w_n(\xi) \neq 0 \) only for \( n = 1 \), \( 2,4 \) or 8

M. F. Atiyah and F. Hirzebruch: “Charakteristische Klassen und Anwendungen” [Characteristic classes and applications], Enseignement Math. (2) 7 : 1 (1961), pp. 188–213. MR 154294 Zbl 0104.39801 article

M. F. Atiyah and F. Hirzebruch: “The Riemann–Roch theorem for analytic embeddings,” Topology 1 : 2 (April–June 1962), pp. 151–166. MR 148084 Zbl 0108.36402 article

In [Borel and Serre 1958] Grothendieck formulated and proved a generalization of the Riemann–Roch theorem which we shall refer to as GRR. This theorem is concerned with a proper morphism \( f:Y\to X \) of algebraic manifolds (any ground field) and reduces to the version (HRR) given in [Hirzebruch 1956] when \( X \) is a point (and the ground field is \( \mathbb{C} \)). It is not known whether GRR or even HRR holds for arbitrary complex manifolds. However the proof of GRR given in [Borel and Serre 1958] breaks up into two separate cases:

\( f \) is an embedding,
\( f \) is a projection \( X\times P_N \to X \), where \( P_N \) is a projective space,

and the main purpose of this paper is to give a proof of GRR in case (i) for arbitrary complex manifolds. This proof is quite different from, and in many ways simpler than, that of [Borel and Serre 1958] and, for the complex algebraic case, it gives a new proof of GRR.

M. F. Atiyah and F. Hirzebruch: “Analytic cycles on complex manifolds,” Topology 1 : 1 (January–March 1962), pp. 25–45. MR 145560 Zbl 0108.36401 article

Let \( X \) be a complex manifold, \( Y \) a closed irreducible \( k \)-dimensional complex analytic subspace of \( X \). Then \( Y \) defines or “carries” a \( 2k \)-dimensional integral homology class \( y \) of \( X \), although the precise definition of \( y \) presents technical difficu1ties. A finite formal linear combination \( \sum n_iY_i \) with \( n_i \) integers and \( Y_i \) as above is called a complex analytic cycle, and the corresponding homology class \( \sum n_iY_i \) is called a complex analytic homology class. If an integral cohomology class \( u \) corresponds under Poincaré duality to a complex analytic homology class we shall say that \( u \) is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class \( u \) satisfies certain topological conditions, independent of the complex structure of \( X \). These conditions are that certain cohomology operations should vanish on \( u \), for example \( \mathrm{Sq}^3u = 0 \): they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.

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Michael F. Atiyah

Papers on K-theory (1959–1970)

Works connected to Friedrich E. P. Hirzebruch

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